We like that. Yeah, Yeah, Yeah. So they are the natural numbers. So there's a few a few letters like this that you'll see. You'll see natural numbers, you'll see integers, you'll see rational numbers, you'll see real numbers. So the natural numbers are like your positive integers. So depending on who you ask, it either starts at zero, it starts at one. And that level of podanticness doesn't really matter, doesn't really kick in until you go off to University. But Yeah, these are the counting numbers goes so one, two, three, four, five, okay, you've then got so z is your integers. So these are both positive and negative and zero. You've got this one that you've probably seen this one the most. They are here. They're the real numbers. So that's basically every number that you can think of that isn't imaginary. You've got q. I've no seeing q before. No. So q is the rational numbers. This is any number. It could be written as an integer divided by an integer. Okay? So so not sds, not pi, not e, but any other number basically. And then you've got A C as well. Actually, you won't seen c because that's further nice, but they're the complex numbers. Oh, okay. Yeah. Okay. In general, you'll see rnz q. Yeah. Okay. There is one specific questions I want to ask. I'll send you a screenshot. Yeah, I'll look it for it in the chat. Oh, 1s way. Sorry, it's on my other, the device call. Yeah. Yeah sorry 1s sorry sorry just it might take like a little wait a little bit time. Is it okay if I can just turn my camera? I'm really sorry because I can't log on to my account on my phone. No. You're wave it at me itbe full screen and so I can read it. Number seven. Yeah. Just pay did not care about q. Okay, it might be easier to read it out and then I'll I'll copy it up. Well, maybe is okay if I just leave the meeting and take a picture of that. Yeah, that works actually. Yeah take a picture of it and then send over. I thought that might be easier actually. Yeah. Okay. I see you in a minute. Oh Yeah, image. Up and on. There, let's go all that wanted. Something out there for me to upload. Always forget how the uploading system works on this. Not there. Where is it? What have I done? Did I do wrong? Why is it not in my drive? Let me try again, save that, save the modrive. There we a sensaved that time. There you go, little wiwrong. The first time it just didn't like it. Okay, so just just number seven. Maybe number five as well. I think if we okay, should we do seven first? Yeah, okay. All why let me back let me crop it. So we do actually and we're gonna need seven okay, so you're given that nasty thing where akm and p and qure constants on, okay, yes, so this this is Oh this is nice, this is so you can work out. The constants sort of one at a time based on the expansion of this. So if we look at the the actual binomia, we've got a minus K X power. I've missed that up. Power m, the power n. Is that right, yeokay? So we can get a straight away quite easily because the first term. When until their constants. Well, what's that power now on the X? I can't read the power. Is it three over two? Three over two? Yes, sorry, that's quite that's right. I've got my glasses on. They're in the other room. Okay. So we know that eighth power n must be but two, four, three over 32. Okay. So that's that's that's your starting point. And then if you look at the second term, so the second term will be a power n minus one times by minus kx power m power one. And this will be times by how many combinations through earth and thatwill always be wherthe power is that I'll be in. Okay then this equals -405 over four x the power three over two. Why do times step by end again? So there's end combination. So you could use, do you like the choose function? Yeah, I like the truth function. So that would be. Nc -13. We am confused now, sorry. So it would be. N choose one or n choose n minus one, which is always the number itself. Is it the same as like is it the same one like n choose one and n choose n minus one? Yes. So that's symmetrical. So it's in the same way that if you had five, choose one or five, choose four, they're the same. Okay, okay. If it's always and it's always in, it's it's always whatever that the power is, if you add five, choose one or five, choose four, that's five. If you've got six, choose one or six, choose five. That's six. Okay? So that's where that ends come from. And then we've got enough info to do anything yet we're only told their constants as well. So they could be positive, they could be negative. Okay, I would personally, I'd just Carry on with this. So I'd then go on my next term got a and minus two times by minus kx m power two times. So this will be nc two, which I don't we don't yet know, doesn't really help us, but this equals 540x per three and then I would start to break down this power. So for example, I can actually already see from that second one, as my pointer sion can see what my mouis. Can you see where my mouth se is? Yeah, Yeah, Yeah. Can you see that this m is the only way we get a power initially. So m must be three over two. You see that about a minus one, so that's on a number. So if that was like that could be like too cubed, so that wouldn't change the power of x. Oh Yeah. Okay, okay. So from that we can go. Okay. Well, m must be 32. And then it's trying to figure out what else we could work out. We could substitute. That's it. How could we do that? Hmm. We could get cube quite nicely from the other end. Okay. Actually we know n as well. We can tell it what n is as well because because this is equal to it's not approximately equal to these are all of the terms. Yeah and then whatever the power is, there's always one extra term and there are power and the power. So you think of a quadratic where it's the power wers two, you always get three times you you get a squared term, you get a linear term, you get a constant. Okay, so from this n must be one, two, three, four, five, so n must be five. Yeah. Okay. From there we then if we know n is five, we can then get a because it's the fifth root of this and then it starts to crumble once you've got that. Oh, that's a very clever it, isn't it? I was just, I was looking for that little trick. Yes, n must be five. So then we can say, Yeah, Yeah, I type that into calculate F A. So what? Fifth route of two, four, three of faoh. These are nice numbers. That's three. That's three over two. I think think so. I don't do a fifth route on this fifth route. There it is. Two, 43Yeah it is so a is three over two. But that's six Marks. That's great. Yeah, I guess I guess p will be the last one we find would be my guess. And then okay, then from there, we know n, we know m, we know a we could then get k because we could go, okay. So this is three over two to the power. I'm looking at this line, three over two to the power. N minus one. That's four times. Minus kx the power three over the two, again, times by five. So what do we get for that? Reto? The power or is 81 over 16? Times by five. We've got 405 over 16. K X three over two. And this equals where we add there. So this equals -405. Over four x the three over two. So now it's just comparing these. So k must be is that minus four? Yeah, thank. Okay Yeah and then we've got k we can then look at. Wow, what's left? What's left after we got k we need p because we found a, we found k, we found a, we found N I, then we could just strobodly, just work up p, right? Yeah, you could then, you could then go for it. Yeah. Just as you've got am and n, we don't actually care what q is. Do we just go? We want the so what what power m will that be? We said n was five. We want the power four. So you just go, okay, so that's a the power one times kx m the power four times by I've choose, for which we know is five. And then plug ging in what we have here. So we got three over two, three over two. And k was, what was k minus four, wasn't it? There it is. So we've got three over two times by minus four x the three over two to the power four. Times by five. So this is what's this going to be? What's minus four power four, 256. So Yeah, over two times, five times, 256, I've got one, nine, 20, x to three of it. Now x to the what's that or two to verse six. Think. So I make P1 thousand nine and 20. Yep, sorry all did you agree? Yeah. Just to be on the safe side, I would also check just to check your values to your m, na and K. I would check to see if it works on one of the other terms. So for example, I would check the I get that with those values. So that would be where n is the power three don't we? So I would, I just go through that and go, okay, so that is a to power two times by kx power, m to power three times. I choose three and I'd go through that process. So we've got three over two squared times minus four x to the three over two cubed. And what is five, two, three? Is that 45? But that's what my brain is saying. Ten miles off. Wait, it's just for these calculators are tyso slowly. Yeah and then they are they are really thing. And then after if I get one thing wrong, my calculation will be wrong. Do you know how to say like a particular like data on this? So I know the easier calculations there's like a stto button can see an stto. So it stands for. Store. Catalogue no memory, not a memory. I don't think that is nlark, is there? There's no store as well. Sdo as well. Okay, we'll have a Google so we can find Yeah. No, that that money was that okay. So if if you plug this in, you do get 1440. Yeah, I checthe 1:21. Yeah, lovely stuff. What's the model of the calculatwe'll? See if we can find it quickly. Sorry, what's the model of that calculator? Ti slash 84 plus ce slash t Python edition slash t Python. How do we do it, saving variables? Hi, know my answer. And the value that is to be stored Press s to no, we said there wasn't an styear, but didn't my let me just look more careful, maybe. Oh bottom m left part one. Bottom left Yeah with the the stto and then an arrow. I might be what why I can't see. So the black buttons to the one above on or is a black button. So enter your value one to do your calculation. Press equals, Press that and then Press alpha and the letter of the variables where you want to store it. And then Press enter. Sorry, repeat that again. Sorry. So get the value you want to be like the answer to the previous calculation. Let's just do some puns name Yeah, and then Press that stoarrow button. Did you Press it? Error, error. Don't Press equals after. That to just Press Press the stto arrow button, then Press alpha. And then the letter or the bottom with the letter you want to store as and then Press enter. Oh yes, yes, I like you. And then they should that they should then stay like indefinitely. How many letters have you got on your one? I know, I know the ones we have do about eight values. I have x or I have so many. Oh my gosh, I have like abc until like the W. Oh, I have it all the way into z. Okay, go. Then when you're doing questions, picking an exam, I would encourage you to write, like if you do store things, just write the letter you have, because obviously there's so many, you might use quite a few throughout the paper therebe a way to like facture reset it as well. So theyprobably make you do that in an exam just to make sure you're not going into with valnot. Not that taking a number room with you is gonna to help you, but there you go. Okay, okay, so we didn't actually need cudid. We no, I guess q was there because we could have worked the other way, we could have worked backwards perhaps but like I don't know why you would can Yeah, Yeah happy with that though. So that's the question. Should should we look at five as well? Does you want five? Yes, please. Thank you. Okay. I'll drag it down. So we've got it down there again. I feel like you're not very good with binthey're a weird why aren't they because you don't you don't see any of it other than like quadratic equations and maybe the odd cubiat jc, it's not really seen so it is is Nope. So you got you got remember you haven't haven't got much experience with it. Like you only started briefly. Okay. We got first two times in the expansion of three plus px. The power Q Q is a natural number, q is Groto five. Then find the coefficient of the x to the five to. Okay. So similar to the last one, you do the expansion algebraically, and then you set up equations with this two 187 and this -10206. Yeah, good. Yeah. We haven't got what button to click. I think I did this during ad birth, but I just can't remember now. It was something and logs. Yeah, Yeah. Wait, the log button. Is catalogue the same as log now? No, I wouldn't have thought so. I can't find love. You got it. No. I think, Oh, no mind, I just tried and narrative and then I think q is seven. Oh, then I'll be fine. Just a nice one. My calcular isn't working for logs for some reason. A lot of the base that I want is it seven first paseven? Yeah, it is seven. There's mc five benefit logs though is if it was a decimal that would have still done it for you. But if you're not, if you're not too Loyet, that's don't worry to four on chapter. Let's have a look. Where is it? I'd have thought itbe somewhere near silcars. Yeah I thought itbe one round there somewhere I if you Press math or vayeah it might be in one mm m. Oh, they give you extra functions, don't they? I just really don't get why they give us these. Like so it's like such sophicated they calculate us because we don't know. It's on the leftmost column of keys directly to the left of the seven key. Left Oh Yeah, leftmost car. Welcome to AI overview. So that will give you base ten log. If you want different bases, you Press the math button. Sorry. Oh wait, no mind. I found it. Yeah, there you go. Yeah log base function. So log base. So Yeah, we want base three and then would want two, two, one, eight, seven. I think it was two, one, seven. Yeah seven th. Okay. And then Yeah, it can see seven. If it was a nasty decimal, then that would still help us. But it wasn't because it's a binome expansion question for twelve, so it wouldn't have been a nasty number. How come when I'm so when I put it on your camera, don't look at the screen whenever I unfoscreen, it always takes away your ability to write it. I know it's so odd. It's just like a system. I guess the second one will be seven. Choose six. Oh, I was trying to go straight for the seven. No, let me I we need p first that way. Yeah. So then Yeah you didn't use the second one for p no pe seven, choose one. Oh, one or six. Yeah, there you go stuff. Plus. Yeah, should it be a plus, sorry, no. Yeah there we go. And that is -10206. Yeah so what is seven times three plus six? Six. I'm seven. Oh, is it minus two? Yeah, I got minus two. Minus. Why is a minus? Because there's a minus here. So it's -10270 here Yeah and then Yeah they the lionce which turns it at once once the extra power five Yeah. Udly, Yeah, we put that minus two in those zo, can't we? So three choose and choose five. Is going to pay. 20 more. 21 times, nine times. Negative six zero, four, eight. Yes. So it should be negative because minus two power five will be a negative. So get a negative is always good. That's what I got as well. And then you just wanted the coefficient. So technically you don't need the x, you wouldn't be marnow for red. Yeah, thank you. And then for can I just exit the meeting really quickly and then do something? Thank you. Again, Yeah do you see the Yeah, I've got it just gone through. Okay. So let's just load in. So I'm going to save that. That's on vectors. So for vvectors, if you want to prove that something's parallel, you prove that like the coefficient is the same. So if a parallel, you need to show that one is a multiple of the other Oh, one. So for example, if it was in terms of like iand js, if you had one I plus one J and five I plus five J, you could factorize out the five and go, Oh, look, they're the same because this is five lots of that. So then they're parallel because they're going in the same direction. What about perpendicular? Now you're asking. So there's a formula evolving cause for the pper pendicular. I don't think thatcome up at year twelve, to be honest. No, I think it's year 13. So the formula, it might be further matactually with a dot product rugie. Oh, let me have a quick look on the textbook. I'm pretty certain it's not year twelve because I didn't teach it last year. Don't think supsuppose the other teacher can of all it vectors. If you find a question that asks you to do that, then we can have a look. But I'm pretty sure it's not not I was just wondering like there is a formula. The formula is where ever I put it. So the formula is. Plus theta equals a dot three over mod a mod b. So a dot b is there's the dot product within done? Dot product? No, it's a dot product. Yeah. I said, I don't think it comes up yet. Dot product is when you do sliwe were doing question five, for example, you want the dot product of a and b, you do five times four plus eight times three. Okay. Then if they're per particular, the angle will be 90. So cos theta would be 90, so this thing would be zero. So they need to have a dot product or. Zero. You know I don't think that comes up yet. No. Oh Yeah. I think it's further mati'm trying to think I'm trying to one is I don't I can't remember. It was all bluring together. Now, which which questions did you want from these ones? I'm the last one. That's number six. That's what we've got. Opq is a triangle. Two lots of pr was pr okay. There it is. Two lots of pr is rq okay. And three lots of or there's three lots of or is os op is A O here is b show that os. It is two a plus b part b the point t is added the diagram such that o to t is minus b the tps last straight line. That is a cool last one. All right. So for let's just move it over, speak a little more space. But we can do that kind. We really need that one bit. Okay. So we need to show os. We want o to s. So we're looking for different routes to get from o to s. So how could you go o to s? 嗯，eo Q Q R R S. You can go op, prr, rs, well, in fact, I think we want to go to straight there. So I would use the fthat they give you here. Okay. So I would start off with what os equals three lots of or and then go, okay, how can I go from o to R? So how could we go from R. One, let me. It says two pr equals q Yeah so of two lots of this is this so these were in the ratio. 12. Aren't they for that one, whereas the other one they've given it in, it's a slightly different way to the three lots of our is os. So they're actually in the same ratio as well, aren't they? They're also two to one. You can say that pq equals. Wait, no sorry, R Q equals. Oh no, sorry, pr equals no. That's right. Wait, yes, that is right. Pr equals a third of a sorry minus b minus a. Yes. So I would write it fully out. But Yeah, so you say that or so however, have you gone from o to R? You went. Okay, nice what you do. Okay, so you've gone and found p to q, haven't you? Okay, so P Q what you say in pq is minus a plus b Yeah around b might say that's the way of writing it. So little p so Peter R, there's a third of that. Yeah what? Okay, then we can use that in our Oh, right. So so R is going to be o to p plus p to R and an op nice and easy today, p to R, we've just worked out a third of b Mara. We can clean that up a bit. What do we get? That third two a plus b. Yep. Yeah you see that and that's or it doesn't want or it once os and we know that os is three lots of or all the way back at the top three so that's two a plus b Yeah so that's three lots of this thing because this two a plus b that we had in the show that so I always found with vectors it's all it's a game of writing loads, you know fully riout every single route that you take first using the capital and then plugging in followers. And I know I know it's really tempted to tithose shortcuts but like even like this, I would I would really for Peter q we should have written out Peter q is Peter a plus o to q just to make it really clear what you've done. And then b so we've got A T so because minus b we know roughly where it is. So it's going to be somewhere about out there. Okay. This is going to be the same distance as manufaccan be a bit further than that, isn't it? That looks a bit too fast about that. Wait, sorry, how do you work that? So because we're given this, so O T is minus b, we know it's from o to t, and then b is a set length. B is always this length. So if we had two b's, it would be twice as long if we had three bthree times long and so on. So because there's only one b, we know it's that same length as that. If it's a positive coefficient, it's going to the left the same direction as if it's a negative it's going to the opposite direction. But it's still parallel because it's still some coefficient times by b, okay? It's just one been in the opposite direction. So I know t already is somewhere over here. Then we need to show that tp and s lie on a straight line. So how do we go about that? Let's move. That's gonna mess that up. That's fine. We need it down here so we can work with it. So how do we show tp and s line a straight line? Pnthird. One tp and really to that they have the same like what called like not multiple but they have the same Yeah Yeah so to prove three points restaurant line you need to prove two things. So this is essentially it's. You had co linear, yes, that's sica. So you pick you pick two vectors made up of these three points. So normally yougo sp and pt, or you could go St and then spp, something like that to show things a collinear, as in they line a straight line to first show they overlap. So that that's easy because we do s to p and p to t. They both include the point p. And then when you share the parallel, okay, if they're parallel untouch, it's a straight line, okay? Just by definition of a straight line. And then to show the parallel is to show that one of them is a multiple of the other, okay? So you think of Luif, you were on like a grid and you went one down and one across. That's not on pen, but one down and one across. You've gone in the same direction as someone that's gone two down and two across. Who's gone in the same direction as someone who's gone three down, three across. You see, look, they're all the different lengths, but they're all in the same direction. Okay? So the same is we can show it for a's and b's. So what do you want to start with? You want to start with sp? Or do you want to start with pt? Actually, youprobably get better off going the other way. Actually thinking about it, I'd do t to p and p to s because they're going to be nicto work with, okay. E two p is. To. Yeah, my writing this. Where you writing this? Oh, I can write it. Yeah, you seem to Birto write a lot smaller than me. Please sorry. So it's is Oh maybe. We wanted t to p. Oh, Yeah, that's fine now. Sorry. Yeah, let's get ahead of myself. The two to b is b plus a. C two p people say Yeah like that Yeah and then we want on P S. So in fact, that shouldn't be too bad actually because with more or less no. In fact, we know pr. No, we do know pr. We wrote that out earlier with ps equals to pr plus rs, Yeah, good. I'm pretty sure pr is up. Pso, we'll scroll back up. Pr, there we go. Pr is a third beam minus, say. Then rs, did we work out? Rs, no, we have got os and we've got our as well. So Yeah, we know our Oh, whereas o is a third two a plus b and we know it's double that space of it. Yeah. So what was oagain? Sorry. Or is that one a third, two a plus b? Two in the bracket. Yeah, there we are, good stuff. And then hopefully, if we clean this up, we get the same amount of baas and a's is. The hope. Okay. And then you wanted some sort of conclusion. How do I prove that it's in the line again? So that's where your conclusion comes into play. So you will say something like vector tp and vector ps share the common point of p and. Tp or youprobably get so sorry, I said that's horble handwriting because I'm writing my iPad so it's just very yes, Yeah because I can read it. Oh really? Oh sorry. Can you please repeat it? Okay, it would be something about so the vectors remember you can use the vector notation. You could just go out that for tp. So Yeah vector tp and vector ps share the point p or both go through p something like that and whatever. So down here, eventually weget ps equals something. In fact, what do we get? Let's have a quick the same length might be be I think they're the same length. Yeah they are right. Next we've got third b plus two thirb. Yeah I think they're both a plus b so pr equals. Rs no ps equals tp like this. But here, look, I think I think they're the same. Okay, Yeah the art is a plus big yes equals tyeah. So that means they're parallel. Sometimes you'll have like a number there. So if you had like that, then that still means the parallel. And so the value of that number doesn't matter. Why are like this equal distance means that they're parallel. So it's not that they're the same length. That means they're parallel, right? So down here, if if this had come out to two a plus two b, then we could say, Oh, well, that means that ps is two lots of a plus b, which means it's two lots of tp. So they're parallel. If you can express one as a vector of another that they're parallel, it's same with that thing. You know if you go one down and one to the right and I go two down and two to the right, even though I've walked a different path, our overall direction is parallel because we've done the same ratio of perpendicular distances. In fact, I don't even need to be perpendicular. Okay? So if you walked a and b and I walked two a and two b, we've gone in the same direction. So the fact that then for the last part of the conclusion, because they are parallel and they touch that must mean they're a straight line. Okay. So sometimes you'll see that called collinear and it's the same conclusion. You just got to point out the obvious that they share a point and then show that two vectors that may cut up, if we wanted to, we could have done the full line so we could have used ts instead of one of the ones we've used. But Yeah you show that one of them is a multiple of the other, therefore they're parallel because they're parallel and they touch there must it must be a straight line. You can't draw two parallel lines that touch that are parallel, that are never a straight light. Just just doesn't make sense. Oh yes. Oh wait I get know Yeah if you have these but two lines like this and then move them to that they're touching they have to be a straight line like it's never going to be that because they're not parallel. Yeah okay, I got it. Okay. Oh my gosh. There's so much stuff Thursday. Oh my God. Fine, right? Now I don't really have any questions, but if recently I have some it's okay by sending in the group chat is all right. Yeah, I can't promise when I'll see them. So don't I'm not on the computer very often. Sento Tuon Monday. Don't worry. I'll tell you how it goes. Yeah, here, please do I want to know and don't you know if it doesn't go as well as you want? Don't worry. That's you know that's that's something to learn from and you know if everydy's else to say it's difficult, then chces are as difficult. If it's if you know if it's if it's a whole year group, if you're struggling, then they've probably given you an absolute stinker of test. I don't know if they're making the test really hard or if like it's just our yogirbest drink like that is so bad at maths, okay, because like last time the applied average was about 40% and it was so bad. Everyone is so bad on that test, okay? But I don't know. It was like our year group is just really bad or it's like, I know it was a really hard test because both I did not have the time to look over like all of the problems because it's just too much. And then also I just don't know how to do some of them. I got I got high than average like still it's not things not good like I just got about 60% which is that's pretty good he's all right yes compared to the average but then like if you but if you take into account of like a level exams Yeah and they can figure that's like that's not very good okay what else think goes this time? Yeah are they it an hour long each Yeah now age well this applied we have the mechanics spwhich I think is better than the Superbad I think so not constant acceleration is it? Forces Yeah forces but there's still a little bit of acceleration. Lina Yeah, I think there's a bit crossover often. Yeah well, we haven't started stats here but that's good. Oh sorry sorry of one last question. Sorry. Really quick for tension. Does tangent equal go to this? Is that right or does it depend on like the the that depends. So if we've got a particle like this and it's got tension here, it's got weight here, if it's accelerating upwards, so if a is going that way, then the resultant force is t minus mg and the equals ma. It's going that way. So it's all about it's all about the acceleration. Yes, if it's going the other way. So if it was going down, then the result force is that. Oh, okay, okay. So you always want to resolve it in the direction that the acceleration is going. Technically, it won't matter. So that if if the acceleration is down and you do this one, all that will happen is you'll get a negative acceleration and the negative results of force and the negative cancbut, you've got to be consistent with them both being positive or both being negative. Okay, easiest way to do that is to always resolve in the direction that this arrow is pointing. If it's pointing up, this one's bigger. Do that one minus the one down or the two down, or have an air down. If it's going down, accelerating down, do the opposite. Okay, thank you so much. Okay, you're very welcome. Good luck. And Yeah I'll see you next week. Let me know how it went. Have I see bye bye.